INCORRECT INVERSE PROBLEM SOLUTION METHOD ......prox i ma tions of the pa ram e ter val ues and the reg u lar iza tion pa ram e ter val - ues have been in ves ti gated. A sta tis ti

INCORRECT INVERSE PROBLEM SOLUTION METHOD FOR PARAMETER IDENTIFICATION OF TRANSPORT

PROCESSES MODELS

by

Emiliya DIMITROVA and Christo BOYADJIEV

Original scientific paperUDC: 532.511:519.6

BIBLID: 0354-9836, 10 (2006), 2, pp. 155-166

A method for model pa ram e ter iden ti fi ca tion on the bases of minimization of the least square func tion has been pro posed. An it er a tive reg u lar iza tionpro ce dure and a nu mer i cal al go rithm have been de vel oped for in cor rect(ill-posed) or es sen tially in cor rect in verse prob lem so lu tion. The methodhas been tested with one and two-pa ram e ter mod els, when the re la tions be -tween ob jec tives func tion and pa ram e ters are lin ear and non-lin ear. The“ex per i men tal” data for pa ram e ters iden ti fi ca tion are ob tained from themodel and a gen er a tor for ran dom num bers. The ef fects of the ini tial ap -prox i ma tions of the pa ram e ter val ues and the reg u lar iza tion pa ram e ter val -ues have been in ves ti gated. A sta tis ti cal ap proach has been pro posed for the anal y sis of the model ad e quacy.It is dem on strated that in the cases of es sen tial in cor rect ness, the leastsquare func tion do not reach min ima. A cri te rion for the in cor rect ness of the in verse prob lem was pro posed.

Key words: model parameter identification, incorrect inverse problems,iterative method, regularization, model adequacy

Introduction

The main prob lem of the mod el ling of the hy dro dy namic, heat and mass trans ferpro cesses is the build-up math e mat i cal struc ture, de scrib ing the pro cesses based on thehy poth e sis (knowl edge) con cern ing to their phys i cal mech a nisms. More over, the pro ce -dure needs of the pa ram e ters iden ti fi ca tion of the math e mat i cal de scrip tion, based on ex -per i men tal data. The in verse iden ti fi ca tion prob lem is of ten an in cor rect (ill-posed), i. e.the so lu tion is sen si ble with re spect to the er rors of the ex per i men tal data [1-5]. The maincause is small pa ram e ters – pre-fac tors of the high-rank ing (sec ond) de riv a tives – in thepar a bolic par tial dif fer en tial equa tions in the hy dro dy namic, heat and mass trans fer mod -els (vis cos ity, diffusivity, con duc tiv ity).

The so lu tion of the pa ram e ters iden ti fi ca tion prob lem can be ob tained by theminimization of the func tional of vari ances (least square func tion), i. e. from the con di -tion for a min i mal dif fer ence be tween cal cu lated and ex per i men tal data [5-10].

There are dif fer ent meth ods (se lec tion, quasi-so lu tion, sub sti tu tion of equa tion)per mit ting to ob tain so lu tions of the in cor rect in verse prob lems in the cases of a pres enceof the ad di tional in for ma tion about the func tional min ima [4, 5, 10].

155

In many cases the in verse prob lems are es sen tial in cor rect and a reg u lar iza tionop er a tor per mits to ob tain the so lu tion [5, 6, 10]. The prob lem reg u lar iza tion usevariational or it er a tive ap proaches. Fur ther the gra di ent meth ods are em ployed for a min -ima search [6, 7, 11-14]. The it er a tive pro ce dure stops when the it er a tive so lu tion movesaway from the ex act so lu tion and the num ber of the last it er a tion is ac cepted as a reg u lar -iza tion pa ram e ter of the in verse prob lem so lu tion.

In many cases, this ap proach gen er ates a large de vi a tion of the it er a tive so lu tionfrom the ex act so lu tion. In the pres ent pa per, the new pro posed it er a tive al go rithm per -mits minimization of the dif fer ence be tween it er a tive and ex act so lu tions.

Problem formulation

Let us con sider a nu mer i cal model:

y = f x b( , )r r

(1)

where f is an objective function, expressed analytically, numerically or through anoperator (algorithm);

rx = (x1, ..., xm) is a vector of independent variables,

rb = (b1, ..., bJ) –

vector of parameters.The pa ram e ters of the model (1) should be de ter mined by means of N ex per i -

men tal val ues of the ob jec tive func tion $ ( $ , ..., $ )y y yN= 1 . This re quires the in tro duc tion ofa least square func tion:

Q y yn nn

N( ) ( $ )rb = -å

=

2

1(2)

where yn n= f x b( , )r r

are the calculated values of the objective function of the model (1),while

rx n = (x1n, ..., xmn) are the values of the independent variables from the different

experimental conditions (regimes), n = 1, ..., N.The pa ram e ters of the model (1) can be de ter mined upon the con di tions im posed

by the min i mum of the func tion Q = (b1, ..., bJ) with re spect to the pa ram e ters rb = (b1, ..., bJ).

The de ter mi na tion of rb faces many trou bles due to the in cor rect ness of the prob -

lem. They are a re sult of the sen si bil ity of the so lu tion with re spect to the ex per i men tal er -rors as so ci ated with the de ter mi na tion of $y. They can be avoided by ap pli ca tions of reg u -lar iza tion meth ods that make the prob lem con di tion ally cor rect.

Incorrectness of the inverse problem

Let us con sider the one-pa ram e ter model:

y = 1 – exp(–bx) (3)

where y is an objective function, x is an independent variable and b is an parameter.In the fig. 1 is shown a de pend ence of the ob jec tive func tion from the model pa -

ram e ter at a con stant value of the in de pend ent vari able x = x0. The re la tion be tween the

156

THERMAL SCIENCE: Vol. 10 (2006), No. 2, pp. 155-166

ob jec tive func tion and the pa ram e ter in the fig. 1 is typ i cal in the cases of small pa ram e ter(vis cos ity, diffusivity, con duc tiv ity) as a pre-fac tor of the sec ond de riv a tive in manymod els of hy dro dy namic, heat and mass trans fer pro cesses.

The fig. 1 per mits to ob tain ob -jec tive func tion y0 for a given pa -ram e ter value b0, i. e. this is the di -rect prob lem so lu tion. The in verseprob lem is an ob tain ing of the pa -ram e ter value b0 if the ex per i men talvalue of the ob jec tive func tion y0 isknown.

Let Dy is an ex per i men tal er rorof the ob jec tive func tion. In the fig.1 is seen, that the er ror of the pa ram -e ter iden ti fi ca tion is dif fer ent forsmall and large ob jec tive func tionval ues. For the small ob jec tive func -tion val ues the er ror Db1 is small and the in verse iden ti fi ca tion prob lem iscor rect. If the ob jec tive func tion val -ues are large, the er ror Db2 is largetoo and the in verse prob lem is in cor -rect (ill-posed). In the case of very large ob jec tive func tion val ues, Db3 is very large andthe in verse iden ti fi ca tion prob lem is es sen tially in cor rect.

The re sults in the fig. 1 show that in verse method in cor rect ness is not re sultof the er ror size and the cause is the pa ram e ter sen si tiv ity with re spect to the ex per i -men tal er rors of the ob jec tive func tion.

Incorrectness of the least square function method

Let us con sider the two-pa ram e ter model:

y = 1 – b1 exp(–b2x) (4)

where b1 1= and b2 5= are exact parameter values.The pa ram e ter iden ti fi ca tion prob lem will be solved by the help of the “ex per i -

men tal” data, ob tained by a gen er a tor of ran dom num bers:

$ ( . . ) , $ ( . . )( ) ( )y A y y A yn n n n n n1 2095 01 09 02= + = + (5)

Here, An are ran dom num bers at the in ter val [0, 1], and yn is ob tained from themodel (5) for x = 0.01n (n = 1,…,100). Ob vi ously, the max i mum rel a tive er rors of the

157

Dimitrova, E., Boyadjiev, Chr.: Incorrect Inverse Problem Solution Method for ...

Figure 1. Objective function y for different valuesof the model parameter b at x = x0 = const.

“ex per i men tal” data ( $)Dy are ±5% and ±10%. The val ues of yn, $( )yn1 and $( )yn

2 are shown infig. 2.

In the fig. 2 is seen that in verse iden ti fi ca tion prob lem is cor rect when 0 < x < 0.3, in -cor rect if 0.31 < x < 0.65, and es sen tially in cor rect when 0.66 < x < 1.

In the figs. 3-5, are seen the hori zon tals of the least square func tion (2) in the cases of±5% rel a tive ex per i men tal data er ror and dif fer ent in ter val of x, when in verse prob lem is

cor rect (fig. 3), in cor rect (fig. 4), and es -sen tially in cor rect (fig. 5). These re sultsshow that the least square method is cor -rect when the dif fer ences be tween ex actpa ram e ter val ues in the model and the co -or di nates of the least square func tionmin i mum are very small (fig. 3). Thesedif fer ences are too large, when the in -verse prob lem is in cor rect (fig. 4). In thecase, when in verse prob lem is es sen tially in cor rect the least square func tion has not a min i mum (fig. 5).

The re sults ob tained show (figs. 3-5), that in the cases of in cor rect in verseprob lems, the least square func tion mini-mization is not lead to so lu tion of the in -verse prob lem and for the prob lem so lu -tion must be use ad di tional con di tions.

158


Figure 2. Mathematical modeland “experimental” data[*] – $( )yn

1 values of y with amaximal “experimental” error of ±5%;[·] – $( )yn

2 values of y with amaximal “experimental” error of ±10%;[—] – y = 1 – exp(–5x)

Figure 3. The horizontals of the least squarefunction Q(n = 1-30; D $y [%] = ±5); [·] – b = [1; 5];

Regularization of the iterative methodfor parameter identification

Var i ous it er a tive meth ods for a min i mum search (the gra di ent ones too) are sta -ble with re spect to the ex per i men tal er rors of the ob jec tive func tions. How ever, af ter cer -tain num ber of it er a tion an in creas ing of the dif fer ence be tween it er a tive and ex act val ues of the pa ram e ters start. That is why in each step must be checked the in creas ing of thisdif fer ence.

The pres ent pa per pro poses a method with a pre lim i nary de fined ac cu racy of thepa ram e ters iden ti fi ca tion. The min i mum of Q(

rb) is de ter mined by a gra di ent method,

con trol ling the dif fer ence be tween it er a tive and ex act pa ram e ter val ues in each it er a tionstep.

Let the it er a tion pro ce dure starts with an ini tial ap prox i ma tion rb ( )0 =

= ( , ..., )( ) ( )b bJ10 0 . The val ues of

rbi = (b1i, ..., bJi), where i is it er a tion num ber, are re sult of

the con di tions im posed by the move ment to wards the anti-gra di ent of the func tion Q(rb):

bji = bj(i–1) – b(i–1)Rj(i–1), j = 1, ... , J (6)

where

R

Q

b

Q

b

j i

j i

j ij

J

( )( )

( )

--

-=

=

æ

è

çç

ö

ø

÷÷

æ

è

çç

ö

ø

÷÷å

11

1

2

1

¶

¶

¶

¶

, , ...,j J=1 (7)

159


Figure 4. The horizontals of the least square function Q(n = 31-65; D $y[%] = ±5); [·] – b = [1; 5];

Figure 5. The horizontals of the least squarefunction Q (n = 66-100; D $y [%] = ±5 ); [·] – b = [1; 5]

Here bi is the it er a tion step and b0 = 10–2 (ar bi trary small step value).Each it er a tion step is suc cess ful if two con di tions are sat is fied:

Q Q y Ri i in

N

n i n i j- -=

- -- = å - -1 11

1 12 2b b( ) ( ) ( )( , ) $f x bfr r ¶

¶

¶

¶

b

Rb

j ij

J

jj

æ

è

çç

ö

ø

÷÷å

é

ë

êê

ù

û

úú

×ì

íï

îï

×æ

è

çç

ö

ø

÷

-=

( )11

f÷å

ü

ýï

þï³

- - - =

-=

- -

( )

( ) ( )( ) ( ) [

ij

J

j i j ji j ib b b b

11

12 2

1

0

2b ( ) ]

, ...,

( ) ( ) ( ) ( )b b R R

j J

j i j i j i j i- - - -- - ³

=

1 1 1 1 0

1

b

(8)

The first con di tion in (8) in di cates that it er a tive so lu tion ( )rb i ap proaches the so -

lu tion at the min i mum ( )*rb , while the sec ond con di tion in (8) con cerns the ap proach of

the it er a tive so lu tion ( )rb i to wards the ex act so lu tion (b). Ob vi ously, it is due to the ef fect

of the prob lem in cor rect ness b ¹rb* (see figs. 3-5).

The re sults ob tained per mit to cre ate an al go rithm for so lu tion of the in verseiden ti fi ca tion prob lems [18, 19].

Correct problem solution

The lit er a ture sources [5, 6, 10, 11], teach that ev ery method for solv ing of in cor -rect prob lems must solv ing a cor rect ones. There fore, the first so lu tion of the in verseprob lem cor re sponds to the in ter val 0 < x < 0.3.

The pro posed al go rithm was used for the so lu tion of the iden ti fi ca tion prob lemand re sults are shown on the tab. 1.

Table 1. One and two-parameter model solutions

D $ [%]y b* i b1* b2

* i

±5 4.9678 337 1.0025 5.0674 128

±10 4.9351 339 0.99401 4.9218 172

Incorrect problem solution

The pa ram e ters iden ti fi ca tion prob lem will be solved by minimization of theleast square func tion (2), where xn = 0.01n, n = 31, ..., 65, i. e. 0.31 £ x £ 0.65.

The in cor rect prob lem so lu tion for the one-pa ram e ter model (b(0) = 6, g = 0.5)and two-pa ram e ter model (b1

0( ) = 1.1, b20( ) = 6, g = 0.05) are shown on the tab. 2.

160


Table 2. Incorrect problem solution

D $ [%]y b* i b1* b2

* i

±5 5.0614 1213 1.1797 5.4666 642

±10 5.1232 1217 1.3778 5.9106 416

The re sults from tab. 2 show that dif fer ences be tween the ex act and the ob tainedval ues of the pa ram e ters are sig nif i cant. The cor rect ness of the pa ram e ter iden ti fi ca tionwill be tested through the model ad e quacy as a criterion [15-17].

Statistical analysis of model adequacy

The model is ad e quate if the vari ance of the ex per i men tal data er ror (Se) isequal to the vari ance of the model er ror (S). The test needs to the ex per i men tal val ues ofthe ob jec tive func tion $ ,yk (k = 1, ..., K) upon iden ti cal tech no log i cal con di tions (re gime) x ==

rx ( )0 = (x xK1

0 0( ) ( ), ... , ), where K =5-10. The ex per i men tal data vari ance re quires es ti ma -tion of the math e mat i cal ex pec ta tion of y( ~ )my [10, 15]:

~ $mK

yy kk

K= å

=

1

1(9)

and as a result

SK

y mk yk

K

e2 2

1

1

1=

--å

=( $ ~ ) (10)

Thus, the vari ance of the model er ror [10] is:

SN J

y yQ

N Jn n

n

N2 2

1

1=

-- =

-å=

( $ ) (11)

where N is experimental data number and J – the parameters number.The model ad e quacy is de fined by the vari ance ra tio:

FS

S=

2

2e

(12)

where S S2 2> e , if S contains the error effect of the both model and experimental data. The value of F is compared to the tabulated values (FJ) of the Fisher’s distribution (criteria)[15]. The condition of the model adequacy is:

F £ FJ (a,n,ne) (13)

where n = N – J, ne = K –1, and a = 0.01-0.1.The sta tis ti cal anal y sis of the one and two pa ram e ters model ad e quacy was

tested for 0 £ x £ 0.30 and the re sults are pre sented on tab. 3. For the test per formed: N =

161


= 30, J = 1(2), K = 10, x(0) = 0.2, and a = 0.05. The re sults col lected con firm the ad e -quacy of the model.

Table 3. Statistical analysis of the model adequacy (0 £ x £ 0.3)

J D $ [%]y b1* b2

* g Se·10–2 S·10–2 F FJ

1 ±5 – 4.9678 0.9 1.7933 1.7071 0.9061 2.24

1 ±10 – 4.9351 0.9 3.5867 3.4139 0.9059 2.24

2 ±5 1.0025 5.0674 0.9 1.7933 1.8354 1.0475 2.25

2 ±10 0.9940 4.9218 0.9 3.5867 3.4434 0.9217 2.25

The sta tis ti cal anal y sis of the cases cor re spond ing to in cor rect in verse prob lem(0.31 £ x £ 0.65) was per formed for N = 35, J = 1(2), K = 10, x(0) = 0.5, and a = 0.05 (seetab. 4). The mod els are ad e quate de spite the large dif fer ences be tween the cal cu lated andthe ex act val ues of the model pa ram e ters (see tab. 2).

Table 4. Statistical analysis of the model adequacy (0.31 £ x £ 0.65)

J D $ [%]y b1* b2

* g Se·10–2 S·10–2 F FJ

1 ±5 5.0614 0.5 2.6042 2.3588 0.8205 2.19

1 ±10 5.1232 0.5 5.2083 4.7328 0.8257 2.19

2 ±5 1.1797 5.4666 0.05 2.6042 2.3656 0.8252 2.20

2 ±10 1.3778 5.9106 0.05 05.2083 4.7349 0.8265 2.20

Essentially incorrect problem

The pa ram e ters iden ti fi ca tion prob lem when in verse prob lem is es sen tially in -cor rect will be solved by minimization of the least square func tion (2), where n = 66, ...,100.

The re sults in the tab. 5 are so lu tions of the iden ti fi ca tion prob lems for one- andtwo-pa ram e ter mod els.

162


Table 5. One and two-parameters model solutions (0.66 £ x £ 1)

D$ [%]y b* i b1* b2

* i

±5 5.1828 2066 2.1720 6.1731 54

±10 5.3816 2156 4.9003 7.4004 128

The re sults in the tab. 5 show that the dif fer ences be tween ob tained and ex actpa ram e ters val ues are very large, but the dif fer ences be tween ob tained and ex act mod elsex hibit just the op po site be hav ior.

Sta tis ti cal anal y sis of the model ad e quacy in the cases of es sen tial in cor rect ness ofthe in verse prob lem (0.66 £ x £ 1) was done for N = 35, J = 1(2), K = 10, x(0) = 0.8, and a =0.05. The re sults are col lected in the tab. 6. The mod els em ployed in this pa per are ad e -quate in de pend ently de spite the large dif fer ences be tween the cal cu lated and the ex actval ues of the model pa ram e ters (see tab. 5).

Table 6. Statistical analysis of the model adequacy (0.66 £ x £ 1)

J D $ [%]y b1* b2

* g Se·10–2 S ·10–2 F FJ

1 ±5 5.1828 5 2.7850 2.5988 0.8707 2.19

1 ±10 5.3816 5 5.5701 5.2482 0.8723 2.19

2 ±5 2.1720 6.1731 5 2.7851 2.6221 0.8855 2.20

2 ±10 4.9003 7.4004 5 5.5701 5.2482 0.8877 2.20

General case

It was shown, that if the “ex per i men tal” data were ob tained upon con di tions (re -gimes), cor re spond ing to the in ter val 0 £ x £ 1, the pa ram e ter iden ti fi ca tion prob lem iscor rect, in cor rect, and es sen tial in cor rect.

In prac tice, very often is pos si ble to have ex per i men tal data in very large re gimein ter val (for ex am ple 0 £ x £ 1). How ever, it is un known which from the ex per i men taldata lead to cor rect or in cor rect prob lem. That is why the pa ram e ter iden ti fi ca tion prob -lem will be solved by minimization of the least square func tion, ob tained in the very large ex per i men tal data in ter val:

Q y yn nn

( ) ( $ )rb = -å

=

2

1

100(14)

where yn = f(xn, rb), xn = 0.01n, n = 1, ..., 100, b = (b1, b2).

In the tab. 7 the re sults of the pa ram e ter iden ti fi ca tion for one- and two-pa ram e -ter model are shown. This re sults are ob tained for ini tial ap prox i ma tion, b(0) = 6, (g = 5)and b1

0( ) = 1.1, b20( ) = 6 (g = 2).

163


Table 7. One and two-parameters model solutions (0 £ x £ 1)

D$ [%]y b* i b1* b2

* i

±5 5.0117 50 1.0106 5.1717 65

±10 5.0231 50 1.0196 5.1721 66

Sta tis ti cal anal y sis of the model ad e quacy in the cases of 0 1£ £x was made forN = 100 and the re sults are shown in the tab. 8. They show that all of mod els are ad e quate due to F < Fj.

Table 8. Statistical analysis of the model adequacy (0 £ x £ 1), K = 10, for differentregimes (x S F x S F x S1

020

300 2 0 5 0 8( ) ( ) ( ). , , ; . , , ; . ,= ¢ ¢ = ¢¢ ¢¢ = ¢¢¢e e e, ¢¢¢F )

b1* b2

* ¢ × -S e 10 2 ¢¢× -S e 10 2 ¢¢¢× -S e 10 2 S·10–2 ¢F ¢¢F ¢¢¢F FJ

– 5.011 1.79 2.60 2.78 2.33 0.80 0.70 1.69 1.99

– 5.023 1.79 1.79 1.79 2.33 1.69 1.69 1.69 1.99

1.010 5.171 1.79 1.79 1.79 2.40 1.80 1.80 1.80 1.99

1.019 5.172 1.79 1.79 1.79 2.40 1.79 1.79 1.79 1.99

Incorrect inverse problem “diagnostics”

In all these cases the dif fer ence be tween cor rect and in cor rect in verse iden ti fi ca -tion prob lem is based on the dis tance be tween ex act so lu tion point and least square func -tion min i mum point. In prac tice how ever the ex act pa ram e ter val ues are un known and acri te rion for the in verse prob lem “di ag nos tics” will be very use ful.

On the tabs. 9 and 10 are shown the so lu tions of cor rect and in cor rect in verseprob lems on the bases of dif fer ent ex per i men tal data sets. It is seen that a cri te rion of thein verse prob lem in cor rect ness is the large dif fer ence be tween so lu tions which are ob -tained on the bases of dif fer ent experimental data sets.

The re sults in the tabs. 9 and 10 show that the so lu tion of the in verse prob lem rb*

per mit to cal cu late the ob jec tive func tion yn n= f x b( , )*r rfor dif fer ent “ex per i men tal”

con di tions (n = 1, …, N). If put this val ues (yn) in eqs. (5), dif fer ent sets of ran dom num -

164


Table 9. Solutions of correct and incorrect problems using different “experimental” data sets

b b10

2011 6( ) ( ). ,= =

Dif fer ent “ex per i men tal” data b1* b2

* g i

0 £ x £ 0.3

1 1.0025 5.0674 0.9 128

2 1.0115 5.1706 0.9 120

3 1.0068 5.1881 0.9 179

0.31 £ x £ 0.65

1 1.1564 5.2675 0.05 798

2 0.5789 3.7056 0.05 1803

3 1.1723 5.2624 0.05 776

bers An per mit to ob tained dif fer ent sets of “ex per i men tal” data $yn (n = 1, …, N). Thecom par i son of the in verse prob lem so lu tions, us ing dif fer ent “ex per i men tal” data sets,will show the in verse prob lem cor rect ness (in cor rect ness).

Table 10. Solutions of essentially incorrect problem and general case, using different “experimental” data sets

b b10

2011 6( ) ( ). ,= =

Dif fer ent “ex per i men tal” data b1* b2

* g i

0.66 < x < 1

1 4.5933 6.7246 5 680

2 0.1161 2.3417 5 390

3 2.7943 5.9219 5 133

0 < x < 1

1 1.0106 5.1716 2 66

2 1.0100 5.1963 2 70

3 1.0134 5.1913 2 76

Conclusions

The pro posed it er a tive method and al go rithm for model pa ram e ters iden ti fi ca -tion in the cases when in verse prob lem is in cor rect shows that a large dif fer ence be tweenpa ram e ter val ues, ob tained on the bases of dif fer ent ex per i men tal data sets, is a cri te rionfor in verse prob lem in cor rect ness.

The so lu tion of the model pa ram e ters iden ti fi ca tion prob lem by the help of theleast square func tion minimization man i fests a large dif fer ence be tween the ex act andcal cu lated (as a func tion min i mum) pa ram e ter val ues i. e. the minimization of the leastsquare func tion is not a so lu tion of the pa ram e ter iden ti fi ca tion prob lem. This dif fer enceis not re sult of the ex per i men tal data size and can be ex plained with the in verse prob lemin cor rect ness, i. e. the pa ram e ter value sen si bil ity with re spect to the ex per i men tal dataer rors.

An ad di tional con di tion is in tro duced for the in verse prob lem reg u lar iza tion,which per mits to use least square func tion minimization for a so lu tion of the model pa -ram e ter iden ti fi ca tion prob lem.

A sta tis ti cal anal y sis of the model ad e quacy is a cri te rion for the ap pli ca bil ity ofthe pre sented it er a tive method for the model pa ram e ters identification.

References

[1] Beck, J. V., Aruold, K. I., Pa ram e ter Es ti ma tion in En gi neer ing and Sci ence, John Wiley andSons, New York, USA, 1977

165


[2] Beck, J. V., Blackwell, B., St. Clair C.R. Jr., In verse Heat Con di tion. III – Posed Prob lems,John Wiley and Sons – Interscience Pub li ca tions, New York, USA, 1985

[3] Glasko, V. B., In verse Prob lem of Math e mat i cal Phys ics, Amer i can In sti tute of Phys icsTrans la tion Se ries, 1988

[4] Tikhonov, A. N., Arsenin, V. I., Meth ods for So lu tions of In cor rect Prob lems (in Rus sian),Nauka, Mos cow, 1986

[5] Tikhonov, A. N, Kal'ner, V. D., Glasko, V. B., Math e mat i cal Mod el ling of Tech no log i calPro cesses and Method of In verse Prob lems (in Rus sian), Mashinostroenie, Mos cow, 1990

[6] Alifanov, O. M., Artiukhin, E. A., Rumiantsev, C. B., Extremal Meth ods for In cor rect Prob -lem So lu tions (in Rus sian), Nauka, Moskow, 1988

[7] Alifanov, O. M., In verse Heat Trans fer Prob lems, Springer-Verlag, Berlin, 1994[8] Brakham, R. L., Jr. Sci en tific Data Anal y sis, Springer-Verlag, Berlin, 1989[9] Banks, H. T., Kunsch, K., Es ti ma tion Tech niques for Dis trib uted Pa ram e ter Sys tems,

Birkhauser, Boston, MA, USA, 1989[10] Boyadjiev, Chr., Fun da men tals of Mod el ing and Sim u la tion in Chem i cal En gi neer ing and

Tech nol ogy (in Bul gar ian), Bulg. Acad. Sci., Inst. Chem. Eng., So fia, 1993[11] Alifanov, O. M., So lu tion of the In verse Heat Con duc tion Prob lems by It er a tive Meth ods, J.

Eng. Phys ics (Rusia), 26 (1974), 4, pp. 682-689[12] Alifanov, O. M., In verse Heat Trans fer Prob lems, J. Eng. Phys ics (Rus sia), 33 (1977), 6, pp.

972-981[13] Alifanov, O. M., On the So lu tion Meth ods of the In verse In cor rect Prob lems, J. Eng. Phys ics

(Rus sia), 45 (1983), 5, pp. 742-752[14] Alifanov, O. M., Rumiantsev, C. B., Reg u lar iza tion It er a tive Al go rithms for the In verse

Prob lem So lu tions of the Heat Con duc tion, J. Eng. Phys ics (Rus sia), 39 (1980), 2, pp.253-252

[15] Draper, N. R., Smith, H., Ap plied Re gres sion Anal y sis, John Wiley and Sons, New York,USA, 1966

[16] Vuchkov, I., Stoyanov, S., Math e mat i cal Mod el ling and Op ti mi za tion of Tech no log i cal Ob -jects (in Bul gar ian), Technics, So fia, 1986

[17] Bojanov, E. S., Vuchkov, I. N., Sta tis ti cal Meth ods for Mod el ling and Op ti mi za tion ofMultifactor Ob ject. (in Bul gar ian), Technics, So fia, 1973

[18] Boyadjiev, Chr., Dimitrova, E., A It er a tive Method for Model Pa ram e ter Iden ti fi ca tion, 1.In cor rect prob lem, Com put ers Chem. Eng., 29 (2005), 4, pp. 941-948

[19] Boyadjiev, Chr., In cor rect In verse Prob lem Con nected with the Pa ram e ter Iden ti fi ca tion ofthe Heat and Mass Trans fer Mod els, Ther mal Sci ence, 6 (2002), 1, pp. 23-28

Author’s address:

Chr. B. Boyadjiev, E. A. DimitrovaBul gar ian Acad emy of Sci ences, In sti tute of Chem i cal EngineeringAcad. G. Bonchev Str., Bl.103, 1113 So fia, BulgariaE-mail: [email protected]

Paper submitted: June 9, 2005Paper revised: May 23, 2006Paper accepted: June 12, 2006

166


Documents

INCORRECT INVERSE PROBLEM SOLUTION METHOD ......prox i ma tions of the pa ram e ter val ues and the reg u lar iza tion pa ram e ter val - ues have been in ves ti gated. A sta tis ti